[adrianmonk]

Here's how I like to think of it.

Floating point numbers are just fractions but with one extra condition: the denominator is a power of two.

The normal rules of fractions apply. If you want to add them, you have to make sure the denominators match, which would involve scaling the numerators too.

Just like fractions, there are multiple ways of writing the same value. 3/2 is the same as 6/4.

You can't write 1/3 exactly because, hey look at your denominator, it's 3. Which isn't a power of 2, is it? So that can't be a floating point value.

[cambalache]

> Just like fractions, there are multiple ways of writing the same value. 3/2 is the same as 6/4.

This is not correct. In binary you only have 2 choices {0,1} for the numerator, so 3/2 is represented 1.1 and 6/4 can only be represented as 1.1 (feel free to add the completing 0s)

[adrianmonk]

The numerator is a binary integer, not a bit (and not the part of the number before the "decimal" point).

3 in binary is 11. 6 in binary is 110.

3/2 (decimal) is 11/10 (binary) and 6/4 (decimal) is 110/100 (binary).

Of course the denominators always have trailing zeros, so can be (and are) represented more compactly.